{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0040",
  "pilot_number": "R0004-02690",
  "number": 2690,
  "family_id": "qf-app21-representation-two-point-rule",
  "title": {
    "zh-Hant": "第 2690 題：數學表徵語義橋",
    "en": "Question 2690: The Nonvertical Affine Rule through Two Distinct Inputs"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r01-mathematical-representation-0040",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-representation"
    ],
    "title_zh": "數學表徵語義橋",
    "stem_zh": "明示文字規則、符號或有限表格。操作=representation-two-point-rule；資料={\"points\":[[2,-4],[5,-13]]}。請依 [points,slope,intercept] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[2, -4], [5, -13]], -3, 2)",
      "explanation_zh": "應用擴展核對=([[2, -4], [5, -13]], -3, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不由圖形外觀補值；有限表徵一致不證未列域上的等價。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "數學表徵語義橋"
      ]
    },
    "variant": {
      "family_id": "qf-app21-representation-two-point-rule",
      "seed": 202610211040,
      "parameters": {
        "points": [
          [
            2,
            -4
          ],
          [
            5,
            -13
          ]
        ],
        "domain": "Finite_representation_bridge",
        "operation": "representation-two-point-rule",
        "semantic_template": "qf-app21-representation-two-point-rule/{\"points\":[[2,-4],[5,-13]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0040",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "29d9e47e9ebaa0e6faae50759f7e7d63584da46d3f7f30767d4223fb8338cb04",
      "family_id": "qf-app21-representation-two-point-rule",
      "seed": 202610211040,
      "semantic_template": "qf-app21-representation-two-point-rule/{\"points\":[[2,-4],[5,-13]]}"
    },
    "fields": {
      "title": "The Nonvertical Affine Rule through Two Distinct Inputs",
      "stem": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, -4], [5, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.",
      "answer": {
        "kind": "expression",
        "value": "([[2, -4], [5, -13]], -3, 2)",
        "explanation": "Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, -4], [5, -13]], -3, 2)."
      },
      "skills": [
        "The Nonvertical Affine Rule through Two Distinct Inputs"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "明示文字規則、符號或有限表格。操作=representation-two-point-rule；資料={\"points\":[[2,-4],[5,-13]]}。請依 [points,slope,intercept] 欄序回答。",
        "en": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, -4], [5, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared symbolic rule or complete finite table",
      "after": "keep the stated domain and representation distinct",
      "relation": "implies",
      "rule_id": "representation_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只由明示規則或表格計算；有限表格的一致性不自動證明未列出論域上的函數等同。",
        "en": "Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "x₂−x₁ := sub(5, 2)",
      "after": "x₂−x₁ = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對兩個橫座標之差非零，才可除。",
        "en": "Check that the two x-coordinates differ before division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x₂−x₁，不以小數近似替代。",
          "en": "Recompute x₂−x₁ with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "2"
        ],
        "output": "3",
        "label": "x₂−x₁"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "y₂−y₁ := sub(-13, -4)",
      "after": "y₂−y₁ = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "縱差使用同一點序，不能只翻轉其中一個差。",
        "en": "Use the same point order in the y-difference;do not reverse just one difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y₂−y₁，不以小數近似替代。",
          "en": "Recompute y₂−y₁ with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-13",
          "-4"
        ],
        "output": "-9",
        "label": "y₂−y₁"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "slope := div(-9, 3)",
      "after": "slope = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由y₂−y₁=a(x₂−x₁)除以非零橫差，求仿射斜率。",
        "en": "Divide y₂−y₁=a(x₂−x₁) by the nonzero x-difference to obtain the affine slope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slope，不以小數近似替代。",
          "en": "Recompute slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-9",
          "3"
        ],
        "output": "-3",
        "label": "slope"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "slope times x₁ := mul(-3, 2)",
      "after": "slope times x₁ = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將斜率代回第一點。",
        "en": "Substitute the slope at the first point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slope times x₁，不以小數近似替代。",
          "en": "Recompute slope times x₁ with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-6",
        "label": "slope times x₁"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "intercept := sub(-4, -6)",
      "after": "intercept = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由y₁=ax₁+b移項求截距。",
        "en": "Rearrange y₁=ax₁+b to obtain the intercept."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 intercept，不以小數近似替代。",
          "en": "Recompute intercept with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "-6"
        ],
        "output": "2",
        "label": "intercept"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "reverse x=2 ax := mul(-3, 2)",
      "after": "reverse x=2 ax = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x=2 ax，不以小數近似替代。",
          "en": "Recompute reverse x=2 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-6",
        "label": "reverse x=2 ax"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "reverse x=2 ax+b := sum(-6, 2)",
      "after": "reverse x=2 ax+b = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x=2 ax+b，不以小數近似替代。",
          "en": "Recompute reverse x=2 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "reverse x=2 ax+b"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "predicted=-4; supplied=-4",
      "after": "both values agree",
      "relation": "implies",
      "rule_id": "two_point_substitution_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反代兩點確認，不把得到的仿射插值誤稱未知曲線的唯一一般公式。",
        "en": "Substitute both points back;the unique affine interpolant is not the unique unrestricted curve through the points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "reverse x=5 ax := mul(-3, 5)",
      "after": "reverse x=5 ax = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x=5 ax，不以小數近似替代。",
          "en": "Recompute reverse x=5 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "5"
        ],
        "output": "-15",
        "label": "reverse x=5 ax"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "reverse x=5 ax+b := sum(-15, 2)",
      "after": "reverse x=5 ax+b = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x=5 ax+b，不以小數近似替代。",
          "en": "Recompute reverse x=5 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-15",
          "2"
        ],
        "output": "-13",
        "label": "reverse x=5 ax+b"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "predicted=-13; supplied=-13",
      "after": "both values agree",
      "relation": "implies",
      "rule_id": "two_point_substitution_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反代兩點確認，不把得到的仿射插值誤稱未知曲線的唯一一般公式。",
        "en": "Substitute both points back;the unique affine interpolant is not the unique unrestricted curve through the points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "two distinct x-coordinates and an affine rule ax+b",
      "after": "unique affine coefficients=(-3,2)",
      "relation": "implies",
      "rule_id": "affine_model_uniqueness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何另一組仿射係數相減後滿足(a−a′)(x₂−x₁)=0，故a=a′且b=b′；若未限制仿射，則可有其他插值。",
        "en": "Subtracting two affine fits gives(a−a′)(x₂−x₁)=0,so a=a′ and b=b′;without the affine restriction,other interpolants are possible."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          2,
          -4
        ],
        [
          5,
          -13
        ]
      ],
      "-3",
      "2"
    ]
  },
  "conclusion": {
    "canonical_value": "([[2, -4], [5, -13]], -3, 2)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/questions.jsonl",
      "batch_sha256": "c3d5564b7f52318668739030fffceffb4febaa2d3d1229aa1ac4a96d58a778b8",
      "original_line": 40,
      "original_line_sha256": "1145ce1182d16c53d1e5ffd3a74c55be597a06baaca5fdbabbb3f6b4cd62772c",
      "question_compact_sha256": "29d9e47e9ebaa0e6faae50759f7e7d63584da46d3f7f30767d4223fb8338cb04",
      "english_row_compact_sha256": "465a1e4e704ce0e151e8c64ff906badf0bfe49038c14d863706c8eadd4f486e2",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.jsonl"
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    "mko_bindings": {
      "mko-mathematical-representation": {
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        "english_sha256": "6774f494e9469b407eedbc3e1c065e7a85ce40b95fbd32d15c7cb728f106fa88",
        "original_path": "public/data/mko/mathematical-representation.json",
        "english_path": "public/data/i18n/editions/mko-mathematical-representation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "564c38c12f40eb905dce200b81cfbefb9e6a554dd6a7a56ba8bb3381039f8b4d",
    "original_candidate_line_sha256": "564c38c12f40eb905dce200b81cfbefb9e6a554dd6a7a56ba8bb3381039f8b4d"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p5",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 13,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9c3aa08f859def5c9b4d04b9bc84c350b34b605ea4542efcc56f88c50858bc21",
      "arithmetic_steps": 9
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}